Coulomb potential, electronic kinetic

This expression is exact within our original approximation, where we have neglected relativistic effects of the electrons and the zero-point motions of the nuclei. The physical interpretation is simple the first term represents the repulsive Coulomb potential between the nuclei, the second the kinetic energy of the electronic cloud, the third the attractive Coulomb potential between the electrons and the nuclei, and the last term the repulsive Coulomb potential between the electrons. 

As indicated, we shall denote electrons and nuclei with Roman (/) and Greek (a) indices, respectively. In terms of kinetic-energy operators for electrons ( e) and nuclei (7k) and the Coulombic potential-energy interactions of electron-electron (Dee), nuclear-nuclear (UNN), and nuclear-electron (VW) type, we can write the supermolecule Hamiltonian as... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The method ofmany-electron Sturmian basis functions is applied to molecules. The basis potential is chosen to be the attractive Coulomb potential of the nuclei in the molecule. When such basis functions are used, the kinetic energy term vanishes from the many-electron secular equation, the matrix representation of the nuclear attraction potential is diagonal, the Slater exponents are automatically optimized, convergence is rapid, and a solution to the many-electron Schrodinger eqeuation, including correlation, is obtained directly, without the use ofthe self-consistent field approximation. 

As we have already indicated, charged particles must be accelerated to kinetic energies on the order of millions of electron volts (MeV) in order to overcome the Coulomb repulsion of another nucleus and induce a nuclear reaction. The Coulomb potential grows with the inverse of the separation between the two ions ... 

The standard quantum chemical model for the molecular hamiltonian Hm contains, besides purely electronic terms, the Coulomb repulsion among the nuclei Vnn and the kinetic energy operator K]. The electronic terms are the electron kinetic energy operator Ke and the electron-electron Coulomb repulsion interaction Vee and interactions of electrons with the nuclei, these latter acting as sources of external (to the electrons) potential designated as Ve]q. The electronic hamiltonian He includes and is defined as... 

Here f is the electron kinetic energy operator, VTe and Vpe are the potential operators for the interactions (coulombic or effective) between the electron and, respectively, the target and the projectile centers. 

Here the quantity U is an effective potential that contains three contributions the kinetic energy for the radial movement of the electrons (in the coordinate a), a centrifugal potential energy, and the Coulomb potential energy — C(a, 12)/R of the system. In the present context of double photoionization it is this Coulomb energy which determines the features of two-electron emission (in atomic units) ... 

This is the single-electron operator including the electron kinetic energy and the potential energy for attraction to the nuclei (for convenience, the single electron is indexed as electron one). The two-electron operators in eq. (2.4) are defined as the Coulomb, J... 

In these expressions written with use of so-called atomic units (elementary charge, electron mass and Planck constant are all equal to unity) RQs stand as previously for the spatial coordinates of the nuclei of atoms composing the system r) s for the spatial coordinates of electrons Mas are the nuclear masses Zas are the nuclear charges in the units of elementary charge. The meaning of the different contributions is as follows Te and Tn are respectively the electronic and nuclear kinetic energy operators, Vne is the operator of the Coulomb potential energy of attraction of electrons to nuclei, Vee is that of repulsion between electrons, and Vnn that of repulsion between the nuclei. Summations over a and ft extend to all nuclei in the (model) system and those over i and j to all electrons in it. 

We start by considering the hydrogen atom, the simplest possible system, in which one electron interacts with a nucleus of unit positive charge. Only two terms are required from the master equation (3.161) in chapter 3, namely, those describing the kinetic energy of the electron and the electron-nuclear Coulomb potential energy. In the space-fixed axes system and SI units these terms are... 

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